An ab initio Langevin dynamics approach is developed based on stochastic density functional theory (sDFT) within a new embedded fragment formalism. The forces on the nuclei generated by sDFT contain a random component natural to Langevin dynamics and its standard deviation is used to estimate the friction term on each atom by satisfying the fluctuation–dissipation relation. The overall approach scales linearly with system size even if the density matrix is not local and is thus applicable to ordered as well as disordered extended systems. We implement the approach for a series of silicon nanocrystals (NCs) of varying size with a diameter of up to 3nm corresponding to Ne = 3000 electrons and generate a set of configurations that are distributed canonically at a fixed temperature, ranging from cryogenic to room temperature. We also analyze the structure properties of the NCs and discuss the reconstruction of the surface geometry.

Quasiparticle (QP) excitations are extremely important for understanding and predicting charge transfer and transport in molecules, nanostructures and extended systems. Since density functional theory (DFT) within Kohn-Sham (KS) formulation does not provide reliable QP energies, a many-body perturbation technique within the GW approximation are essential. The steep computational scaling of GW prohibits its use in extended, open boundary, systems with thousands of electrons and more. Recently, a stochastic formulation of GW has been proposed [Phys. Rev. Lett. 113, 076402 (2014)] which scales nearly linearly with the system size, as illustrated for a series of silicon nanocrystals exceeding 3000 electrons. Here, we implement the stochastic GW (sGW) approach to study the ionization potential (IP) of a subset of molecules taken from the "GW 100" benchmark. We show that sGW provides a reliable results in comparison to GW WEST code and to experimental results, numerically establishing its validity. For completeness, we also provide a detailed review of sGW and a summary of the numerical algorithm.

The second-order Matsubara Green’s function method (GF2) is a robust temperature dependent quantum chemistry approach, extending beyond the random-phase approximation. However, till now the scope of GF2 applications was quite limited as they require computer resources which rise steeply with system size. In each step of the self-consistent GF2 calculation there are two parts: the estimation of the self-energy from the previous step’s Green’s function, and updating the Green’s function from the self-energy. The first part formally scales as the fifth power of the system size while the second has a much gentler cubic scaling. Here, we develop a stochastic approach to GF2 (sGF2) which reduces the fifth power scaling of the first step to merely quadratic, leaving the overall sGF2 scaling as cubic. We apply the method to linear hydrogen chains containing up to 1000 electrons, showing that the approach is numerically stable, efficient and accurate. The stochastic errors are very small, of the order of 0.1% or less of the correlation energy for large systems, with only a moderate computational effort. The first iteration of GF2 is an MP2 calculation that is done in linear scaling, hence we obtain an extremely fast stochastic MP2 (sMP2) method as a by-product. While here we consider finite systems with large band gaps where at low temperatures effects are negligible, the sGF2 formalism is temperature dependent and general and can be applied to finite or periodic systems with small gaps at finite temperatures.